Problem 61 - Algebra 2

Question

PREREQUISITE SKILL Find the mean, median, mode, and range for each set of data. Round to the nearest hundredth, if necessary. (Pages 759 and 760 ) $$ 80,50,65,55,70,65,75,50 $$

Step-by-Step Solution

Answer

Mean: 63.75, Median: 65, Mode: 50 and 65, Range: 30.

Step 1: Arrange the Data

First, we need to organize the numbers in ascending order to make calculations easier. The provided dataset is 80, 50, 65, 55, 70, 65, 75, 50. Reordering these numbers gives us: 50, 50, 55, 65, 65, 70, 75, 80.

Step 2: Calculate the Mean

The mean is calculated by summing all the numbers and dividing by the count of numbers. Sum: 50 + 50 + 55 + 65 + 65 + 70 + 75 + 80 = 510. The count is 8, as there are 8 numbers. So, the mean is \(\frac{510}{8} = 63.75\).

Step 3: Calculate the Median

For the median, first make sure the numbers are ordered, which we've done in Step 1. Since there are 8 numbers (an even set), the median will be the average of the 4th and 5th numbers. These numbers are 65 and 65, so the median is \(\frac{65+65}{2} = 65\).

Step 4: Identify the Mode

The mode is the number that appears most frequently in the dataset. In our ordered list, 65 appears twice, and so does 50, more than any other number. Thus, this dataset is bimodal with modes 50 and 65.

Step 5: Calculate the Range

The range is the difference between the highest and the lowest numbers in the dataset. Here, the highest number is 80, and the lowest number is 50. Thus, the range is \(80 - 50 = 30\).

Key Concepts

Mean

The mean, also known as the arithmetic average, represents the sum of numbers divided by the count of numbers. It provides a central value that reflects the overall distribution of a data set. To find the mean:

• Add all the numbers together to get the total sum.
• Count how many numbers are in the set.
• Divide the total sum by the count of numbers.

In the example given, we add the numbers 50, 50, 55, 65, 65, 70, 75, and 80, resulting in a sum of 510. Since there are 8 numbers, we divide 510 by 8, giving us a mean of 63.75. The mean helps us understand the average or typical value one might expect in a set of data.

Median

The median is the middle value in an ordered dataset, providing another view of the central tendency. Arranging the data in ascending order is crucial here. The purpose of the median is to identify the center of a dataset, particularly useful when dealing with outliers.

• First, order the numbers from least to greatest.
• If there is an odd number of data points, the median is the middle number.
• If there is an even number of data points, calculate the average of the two middle numbers.

In our data set of 50, 50, 55, 65, 65, 70, 75, and 80, since there are 8 numbers, we take the middle pair, 65 and 65, and find their average, which results in a median of 65. The median gives a good indicator of the dataset's midpoint without being skewed by extreme values.

Mode

The mode is the value that appears most frequently in a dataset. It highlights the most common occurrence within the set. Identifying the mode is useful in situations where the most common item is of interest. Here's how to find the mode:

• Count how many times each number appears.
• The number that appears the most is the mode.

In our example, the numbers 50 and 65 each occur twice, which is more than any other numbers, making this dataset bimodal with the modes being 50 and 65. Having more than one mode is possible and signifies multiple common points in the data distribution.

Range

The range represents the span between the highest and lowest values in a dataset. It provides insight into the spread or variability of the numbers. A broader range indicates more variability, while a narrower range indicates less.

• Identify the smallest and largest numbers in the set.
• Subtract the smallest number from the largest one.

In the given dataset, the smallest number is 50 and the largest is 80. Therefore, the range is calculated as 80 minus 50, which equals a range of 30. Understanding the range helps in assessing how spread out the data values are and can indicate potential outliers or variability in the dataset.